Musings

I started off by recounting the tale of Howard Georgi, back in 1982, warning me off studying quantum gravity, as a waste of time. The point is that there’s no decoupling regime in which quantum “pure gravity” effects are important, while other particle interactions can be neglected. “Universality” in field theory — usually our friend — is, here, our enemy. Unless we know all particle physics interactions all the way from accessible energy up to the Planck scale, we can never hope to extract any quantitative predictions about quantum gravitational effects.

What about treating gravity classically? Maybe we don’t need to quantize it. Matter is certainly quantum mechanical. But, perhaps, we could just treat Einstein’s equation classically. Of course, if matter is quantum-mechanical, then $T_{\mu\nu}$ is an operator on Hilbert space, not a c-number. To make sense of the Einstein equation as a classical equation, we should presumably replace $T_{\mu\nu}$ by its expectation value, $\langle T_{\mu\nu}\rangle$.

The trouble is that $\langle T_{\mu\nu}\rangle$ depends nonlinearly on the state of the matter system. If we then solve the Einstein equation for the metric, and plug back in to contruct the time-evolution operator of the matter system, we discover that time-evolution is no longer given by a linear operator on Hilbert space. This is a disaster! People have tried to contruct nonlinear modifications of quantum mechanics. All such attempts have failed miserably.

Still, we manage, for the most part, to live happy, fulfilled lives while ignoring this colossal failure. As long as we can neglect the back-reaction of the matter system on the metric (i.e., as long as we do not attempt to solve the Einstein equation), we can perfectly well do quantum field theory in a fixed gravitational background. And we can happily do general relativity, provided we are willing to ignore the quantum mechanical nature of matter, i.e. to treat $T_{\mu\nu}$ as a classical c-number field.

That, however, won’t cut it if we want to study the quantum mechanics of blackholes, or the physics of the very early universe (for instance).

It’s often said that it is difficult to reconcile quantum mechanics (quantum field theory) and general relativity. That is wrong. We have what is, for many purposes, a perfectly good effective field theory description of quantum gravity. It is governed by a Lagrangian

This is a theory with an infinite number of coupling constants (the $c_i$ and, all-importantly, the couplings in $\mathcal{L}_{\text{matter}}$). Nonetheless, at low energies, i.e., for $\varepsilon\equiv \frac{E^2}{M_{pl}^2}\ll 1$, we have a controllable expansion in powers of $\varepsilon$. To any finite order in that expansion, only a finite number of couplings contribute to the amplitude for some physical process. We have a finite number of experiments to do, to measure the values of those couplings. After that, everything else is a prediction.

In other words, as an effective field theory, gravity is no worse, nor better, than any other of the effective field theories we know and love.

The trouble is that all hell breaks loose for $\varepsilon\sim 1$. Then all of these infinite number of coupling become equally important, and we lose control, both computationally and conceptually.

An analogy with a more familiar case is helpful. Fermi theory (where we augment the original charged current interactions with neutral current ones, extend it beyond the first generation, …) is a perfectly adequate low-energy effective Lagrangian of the weak interactions. One has, schematically, a 4-Fermi interaction (a dimension-6 operator), $$\frac{1}{M^2} \overline{\psi}\gamma_\mu P_L\psi \overline{\psi}\gamma^\mu P_L \psi$$ plus the usual panoply of yet-higher-dimension operators, suppressed by higher powers of $M\sim 100$ GeV. For $\varepsilon=\frac{E^2}{M^2}\ll 1$, this is a perfectly valid effective field theory, which adequately described particle physics until 1983 (when the W and Z bosons were first produced experimentally). For $\varepsilon \sim 1$, it need to be replaced by some other theory, involving new degrees of freedom. Glashow, Weinberg and Salam won their Nobel prize in 1979 — i.e., well before the breakdown of Fermi theory — for proposing its successor.

So, one possibility for solving the problem of the breakdown of (1) is that new degrees of freedom appear at the Planck scale. But it’s not obvious what sort of degrees of freedom would come to our rescue here (not to mention the already uncomfortable fact that we don’t know what the degrees of freedom in $\mathcal{L}_{\text{matter}}$ should be, even below the Planck scale).

Another possibility is that the appearance of an infinite number of independent couplings is an illusion. Perhaps the physics of (1) is actually controlled by a UV fixed point. At the fixed point, the infinite number of couplings, rather than being independent, are actually functions of a small number of parameters, coordinatizing the fixed point set.

That’s a very attractive idea. Note that, unlike the previous case, where introducing new degrees of freedom might conceivably yield a theory which is weakly-coupled, the UV fixed-point theory is, almost certainly, going to be strongly-coupled, and so will need to be treated by nonperturbative methods.

There are various folks who purport to be attempting to “nonperturbatively” quantize general relativity. They are, secretly (whether they know it or not) pinning their hopes on the existence of a suitable UV fixed point. Otherwise, their theory has an infinite number of independently-adjustable couplings (and is, therefore, utterly unpredictive¹). They may not have bothered to inquire about the number of independent couplings in their model but, barring a UV fixed point, you run into this problem, however studiously you avoid confronting it.

Moreover, if you’re pinning your hopes on a UV fixed point, Georgi’s objection now comes back to bite you. If you, somehow, managed to find a UV fixed point, and then you diddle with the matter content of your theory, you will end up totally missing the erstwhile UV fixed point. Adding matter to your theory cannot be an afterthought (as most of those hoping to “quantize GR” assume). You need to get your matter content right from the git-go. As Howard pointed out, that’s essentially impossible …

Escape

So, how does string theory beat these seemingly insurmountable odds?

It provide a unique, or nearly unique UV completion, not by having a fixed point, but by having a tower of higher-spin gauge symmetries that constrain the seemingly independent couplings of (1). In fact, they are so tightly constrained that there aren’t any continuously-adjustable coupling constants at all!

These higher spin gauge symmetries must be spontaneously-broken, and hence the corresponding higher spin “gauge fields” are massive (like the W’s and Z). But, even there, the existence of a consistent interacting theory of higher spin fields is only possible if they are infinite in number.

This is a funny trade-off. Rather than a finite number of (undetermined) UV degrees of freedom, with (undetermined) interactions, we have an infinite number. But their spectrum is completely constrained, and their interactions determined by the Ward identities associated to these broken higher spin gauge symmetries.

(One residuum of this is that, at very high energies, these higher spin gauge symmetries are effectively restored. The Coleman-Mandula Theorem then insures that the S-matrix is trivial. And, indeed, string scattering is very soft at high energies.)

Observables

Another conundrum of quantum gravity is what are the observables?

We know that proper observables must be gauge-invariant (i.e. diffeomorphism-invariant). In theories with asymptotic regions, you need to be a bit careful. You should only mod out by those gauge transformations which go to the identity at infinity. Those which act nontrivially at infinity are global symmetries of your theory (rather than gauge redundancies). Hence the existence of global charge in QED.

We only know of a few examples where the complete set of observables of quantum gravity have been determined. The answer depends very much on the asymptotics of our spacetime.

1. In asymptotically flat space, the group of residual diffeomorphisms is Poincaré. When disturbances are widely-separated, they are essentially non-interacting. So we start with some non-interacting degrees of freedom, classified by representations of Poincaré, in the far past; we end up with a similar set on non-interacting degrees of freedom in the far future. And we’re interested in the transformation from the “in” state to the “out” state. In other words, the observables of the theory are S-matrix elements.
2. In asymptotically anti-de Sitter space, the residual diffeomorphisms are the anti-de Sitter group, which is isomorphic to the conformal group in one lower dimension. Maldacena showed that the observables are the Green’s functions of a (conformal) quantum field theory on the boundary of our asymptotically AdS space.

These are very different-looking answers. No one knows, for instance, what the complete set of observables is, in the case of compact spatial topology. And you’re not going to be able to hazard a guess, based on these two examples.

Background Independence

I didn’t get to discuss this in class, and I think I’m not going to try to here. It’s been discussed to death at the String Coffee Table and on Cosmic Variance. No point in wasting more electrons rehashing those arguments.

Update: Discretized Theories

Since Wolfgang asks below, and since they seem to be the subject of rampant confusion, let me add a few, very general, remarks on discretized theories. Say we introduce a discretization, which thereby endows our theory with a fundamental length scale, $l$. At distance scales much longer than $l$, one expects to recover an effective continuum Lagrangian, of the general form (1), with coefficients which depend on the couplings of the discretized theory and on $l$.

Naïvely, one expects $M_{pl}^2\sim 1/l^2$. The “continuum limit” is obtained by taking $l\to 0$, while holding $M_{pl}$ fixed. That requires a fine-tuning of the couplings of the discretized theory. The required fine-tuning is awkward and, apparently, very difficult to achieve in practice. So some people have gotten the bright idea to stop trying, and simply leave $l$ fixed, rather than taking it to zero.

That would have made life much easier, had I not maliciously suppressed an important feature of (1). Namely, there’s a cosmological constant term, $$\Delta S = \Lambda \int d^4 x \sqrt{-g}$$ and, naïvely, $\Lambda \sim 1/l^4$. Even if we give up on give up on taking the continuum limt, so as to avoid the fine-tuning required to keep $M_{pl}^2\ll 1/l^2$, we still have to fine-tune in order to obtain $\Lambda\ll 1/l^4$.

Urs is right that LQG isn’t, strictly, a discretized model, though the use of the spin-network basis does introduce a fundamental length scale into the theory. It’s, more properly, a continuum theory, quantized in a Hamiltonian framework (albeit, a very, very unconventional one). The words I wrote above were geared to a Lagrangian formalism. It’s not hard to adapt them to a Hamiltonian one.